A mass m=4kg is attached to both a spring with spring constant k=577N/m and a dash-pot with damping constant c=4N⋅s/m . The mass is started in motion with initial position x0=5m and initial velocity v0=7m/s . Determine the position function x(t) in meters.

Explanation:

The position and velocity functions have the following forms:

Where:

First, the angular frequency of oscillation is calculated:

Later, it is needed to determine if system is underdamped, critically damped or overdamped. Critic value is given by . The value is:

As c is lower than α, the system has an underdamped behavior. The initial values for position and velocity are, respectively:

Then,

By making some algebraic handling:

The amplitude is:

The position function is:

the answer would be a

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west